Double covering of curves

Let $C$ be a smooth projective algebraic curve of genus $q$ and $g$ an integer with $g\ge 4q+5$. For all integers $d\ge g-2q+1$ we prove the existence of a double covering $f:X\to C$with $X$ a smooth curve of genus $g$ and the existence of a degree $d$ morphism $u:X\to \mathbb P^1$ that does not factor through $f$. By the Castelnuovo-Severi inequality, the result is sharp (except perhaps the bound $g\ge 4q+5$).